WP 2.8: add
define('FS_METHOD', 'direct');
define('WP_TEMP_DIR', ABSPATH.'wp-content/tmp');
in wp-config.php
The function in file.php reads FS_METHOD and the tmp file creation test is never executed.
Make sure the wp-content/tmp directory exists and is writable by the apache user

Pre WP 2.8: well... you may need to patch the function in file.php by removing the if statement before $method='direct';
see comment #38 above
Only do this if you're sure you've setup file ownership correctly.

BTW the three numbers in chmod describe rights for "user", "group" and "others". read==4, write==2, execute==1. Sum of rights make up the number. Hence 777 means read, write and execute for user, group and others. 660 means read and write for user and group, no rights for others.

NEVER do a chmod 777 on any file or directory. NEVER recommend anyone to do it either!